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The following stochastic differential equation describes the behaviour of the price St, t0, of a share. dSt = (a+bsin(t)) Stdt + σS₁dWt, where a, b, σ are constants such that a > 0, |b| < a, and σ > 0. (a) Compute the differential of the function F(t) defined by F(t) = ln(St). (b) Solve the equation for St with the initial value S(0) = So. (c) Compute E(In(St)) and Var(ln(St)) and state the distribution of In(St).

The following stochastic differential equation describes the behaviour of the price St, t0, of a share. dSt = (a+bsin(t)) Stdt + σS₁dWt, where a, b, σ are constants such that a > 0, |b| < a, and σ > 0. (a) Compute the differential of the function F(t) defined by F(t) = ln(St). (b) Solve the equation for St with the initial value S(0) = So. (c) Compute E(In(St)) and Var(ln(St)) and state the distribution of In(St).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The following stochastic differential equation describes the behaviour of the price St, t0, of a share. dSt = (a+bsin(t)) Stdt + σS₁dWt, where a, b, σ are constants such that a > 0, |b| < a, and σ > 0. (a) Compute the differential of the function F(t) defined by F(t) = ln(St). (b) Solve the equation for St with the initial value S(0) = So. (c) Compute E(In(St)) and Var(ln(St)) and state the distribution of In(St).
Transcribed Image Text:The following stochastic differential equation describes the behaviour of the price St, t0, of a share. dSt = (a+bsin(t)) Stdt + σS₁dWt, where a, b, σ are constants such that a > 0, |b| < a, and σ > 0. (a) Compute the differential of the function F(t) defined by F(t) = ln(St). (b) Solve the equation for St with the initial value S(0) = So. (c) Compute E(In(St)) and Var(ln(St)) and state the distribution of In(St).
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